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The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

https://doi.org/10.1007/s00526-021-01938-2

Banerjee, Agnid ; Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ( April 2021 , Calculus of Variations and Partial Differential Equations)

Abstract

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight$$|y|^a$$ ${| y |}^{a}$ for$$a \in (-1,1)$$ $a \in (- 1, 1)$ . Such problem arises as the local extension of the obstacle problem for the fractional heat operator$$(\partial _t - \Delta _x)^s$$ ${(\partial_{t} - Δ_{x})}^{s}$ for$$s \in (0,1)$$ $s \in (0, 1)$ . Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ $a = 0$ ).